Bayesian Hypothesis Testing for Gaussian Graphical Models:Conditional Independence and Order Constraints

Abstract

<p>Gaussian graphical models (GGM; partial correlation networks) have become increasingly popular in the social and behavioral sciences for studying conditional (in)dependencies between variables. In this work, we introduce exploratory and confirmatory Bayesian tests for partial correlations. For the former, we first extend the customary GGM formulation that focuses on conditional dependence to also consider the null hypothesis of conditional independence for each partial correlation. Here a novel testing strategy is introduced that can provide evidence for a null, negative, or positive effect. We then introduce a test for hypotheses with order constraints on partial correlations. This allows for testing theoretical and clinical expectations in GGMs. The novel matrix$-F$ prior distribution is described that provides increased flexibility in specification compared to the Wishart prior. The methods are applied to PTSD symptoms. In several applications, we demonstrate how the exploratory and confirmatory approaches can work in tandem: hypotheses are formulated from an initial analysis and then tested in an independent dataset. The methodology is implemented in the R package BGGM.</p>

Quantifying the Reliability and Replicability of Psychopathology Network Characteristics

“…It would be valuable for future research to continue to develop methods for comparing a wider variety of the properties that characterize networks and their structural differences, which is an ongoing challenge for the quantification of differences between networks (Schieber et al, 2017). Another promising direction for comparing estimated network structures is in the emergence of methods for Bayesian hypothesis testing in GGMs (Williams & Mulder, 2019). Estimating networks in this framework would facilitate confirmatory testing of models consistent with network theory, as well as the comparison of competing theoretical models in real data, representing an important step forward for the field.…”

BGGM: A R Package for Bayesian Gaussian Graphical Models

“…The following methods were introduced in Williams and Mulder (2019). That work not only presented an exploratory approach using the Bayes factor, but it also proposed methodology for confirmatory hypothesis testing in GGMs.…”

“…It overcomes one limitation of the Wishart prior (Section 3.1). Namely, as shown in Williams and Mulder (2019), the prior standard deviation is necessarily a function of the network size (i.e., p −1/2 ). This limits its usefulness for Bayesian hypothesis testing in particular.…”

“…These include Bayesian approaches for assessing in-and out-of-sample predictive accuracy, testing for differences between partial correlations, and importantly, confirmatory hypothesis testing with the Bayes factor. This allows for rigorously testing scientific theories or expectations in GGMs, as well as explicitly gaining evidence for the null hypothesis of conditional independence (Williams and Mulder 2019). Further, BGGM includes two approaches for comparing any number of GGMs (Williams et al 2019a) -e.g., those estimated from different groups (male vs. female).…”

Bayesian Estimation for Gaussian Graphical Models: Structure Learning, Predictability, and Network Comparisons

“…The methods in the work belong to the former, and thus, the prior distribution plays little role in inference. With colleagues, I have developed Bayesian methodology for hypothesis testing in GGMs (Williams & Mulder, 2019a;Williams, Rast, Pericchi, & Mulder, 2019). These methods are also implemented in the package BGGM.…”